Measuring out-of-time-order correlators on a quantum computer based on an irreversibility-susceptibility method

TL;DR

This work addresses measuring out-of-time-ordered correlators (OTOCs) on near-term quantum hardware by comparing three protocols—rewinding time method (RTM), weak-measurement method (WMM), and irreversibility-susceptibility method (ISM)—in a 4-qubit XXZ spin chain prepared in a finite-temperature Gibbs state. It provides the first experimental demonstration of ISM on the Quantinuum reimei emulator and analyzes method-dependent performance against ideal and noiseless simulations. The results show overall agreement with theory, while revealing distinct, method-specific deviations that depend on dynamics and Hamiltonian parameters, highlighting practical considerations for experimental scrambling studies. Collectively, the work validates multiple practical routes to probe quantum chaos on near-term devices and outlines concrete directions for scaling, state-preparation fidelity, and error mitigation to enable quantitative OTOC measurements. The findings have implications for benchmarking quantum simulators and for understanding scrambling in realistic quantum many-body systems.

Abstract

The out-of-time-ordered correlator (OTOC) is a powerful tool for probing quantum information scrambling, a fundamental process by which local information spreads irreversibly throughout a quantum many-body system. Experimentally measuring the OTOC, however, is notoriously challenging due to the need for time-reversed evolution. Here, we present an experimental evaluation of the OTOC on a quantum computer, using three distinct protocols to address this challenge: the rewinding time method (RTM), the weak-measurement method (WMM), and the irreversibility-susceptibility method (ISM). Our experiments investigate the quantum dynamics of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. As a key contribution, we provide the first experimental demonstration of the ISM, using the numerical emulator of the trapped-ion quantum computer, reimei. We also conduct a detailed comparative analysis of all three methods, revealing method-dependent behaviors in the measured OTOC. This work not only validates these protocols as practical tools for exploring quantum chaos on near-term hardware but also offers crucial insights into their respective advantages and limitations, providing a practical framework for future experimental investigations.

Figures (4)

• Figure 1: Schematic diagram of the rewinding time method (RTM). The RTM is a procedure to evaluate the four-point correlator $F_{\beta}(\tau)$. The process begins with a controlled operation $| {0} \rangle\langle {0} |_{\mathbf{C}}\otimes\openone_{\mathbf{S}}+| {1} \rangle\langle {1} |_{\mathbf{C}}\otimes V(0)_{\mathbf{S}}$. This is followed by a forward time evolution $\openone_{\mathbf{C}}\otimes U(\tau)_{\mathbf{S}}$, an operation $\openone_{\mathbf{C}}\otimes W_{\mathbf{S}}(0)$, and a backward time evolution $\openone_{\mathbf{C}}\otimes U^{\dagger}(\tau)_{\mathbf{S}}$. The sequence concludes with a final controlled operation $| {0} \rangle\langle {0} |_{\mathbf{C}}\otimes V(0)_{\mathbf{S}}+| {1} \rangle\langle {1} |_{\mathbf{C}}\otimes\openone_{\mathbf{S}}$. Finally, the control system $\mathbf{C}$ is measured in both the $\sigma^{x}$ and $\sigma^{y}$ bases to calculate $F_{\beta}(\tau)$.
• Figure 2: Schematic diagram of the weak-measurement method (WMM). The WMM is a sequence of operations designed to evaluate the squared commutator $C_{\beta}(\tau)$. The protocol consists of alternating weak measurements and time evolutions: A weak measurement $M^{V}_{v}(\phi_{v})$ via the weak interaction $S_{V}(\phi_{v})=\exp\left[-\mathrm{i}\phi_{v}(V\otimes Y)/2\right]$, The forward time evolution $U(\tau)$, A second weak measurement $M^{W}_{w}(\phi_{w})$ using the weak interaction $S_{W}(\phi_{w})=\exp\left[-\mathrm{i}\phi_{w}(W\otimes Y)/2\right]$. Apply the backward time evolution $U^{\dagger}(\tau)$, a third weak measurement $M^{V}_{v'}(\phi_{v'})$ via the weak interaction $S_{V}(\phi_{v'})=\exp\left[-\mathrm{i}\phi_{v'}(V\otimes Y)/2\right]$, the forward time evolution $U(\tau)$, and a fourth weak measurement $M^{W}_{w'}(\phi_{w'})$ using the weak interaction $S_{W}(\phi_{w'})=\exp\left[-\mathrm{i}\phi_{w'}(W\otimes Y)/2\right]$. Finally, $C_{\beta}(\tau)$ is determined by averaging the products of the modified eigenvalues $\alpha_{v}(\phi_{v}),\alpha_{w}(\phi_{w}),\alpha_{v'}(\phi_{v'}),\alpha_{w'}(\phi_{w'})$ obtained from these measurements.
• Figure 3: Schematic diagram of the irreversibility-susceptibility method (ISM). The ISM is a protocol designed o evaluate the squared commutator $C_{\beta}(\tau)$ and consists of a sequence of operations: A weak interaction $U_{V,\theta}=\exp\left[-\mathrm{i}\theta(Z\otimes V)\right]$, a scrambling process $\mathcal{D}_{W}(\bullet):=W(\tau)(\bullet)W^{\dagger}(\tau)$, where $W(\tau)=U^{\dagger}(\tau)W(0)U(\tau)$ from $t=0$ to $t=\tau$, and a recovery map $\mathcal{R}_{V,\mathbf{S}}:=\mathcal{J}_{\mathbf{S}}\circ\mathcal{U}^{\dagger}_{V,\theta}$, which is given by the inverse process of the weak interaction $\mathcal{U}_{V,\theta}$ and $\mathcal{J}_{\mathbf{S}}(\bullet):=\sum_{j=\pm}\bra{j}\mathrm{Tr}_{\mathbf{S}}(\bullet)\ket{j}| {j} \rangle\langle {j} |_{\mathbf{Q}}$. Finally, the irreversibility of the entire process is calculated by comparing the initial and final states of the ancilla qubit system $\mathbf{Q}$.
• Figure 4: The experimental results of the measurement of the OTOC. The results present a comparison among the ideal values (solid lines) from matrix calculations, the noiseless values (circle dots) from aer-simulator, and the experimental results (square dots) from the reimei emulator. The error bars represent the standard deviation of the measured values. While the experimental data show good overall agreement with the theoretical predictions, clear method-dependent behaviors of the OTOC are observed. (a) is the OTOC evaluated using the rewinding time method (RTM). (b) is the OTOC evaluated using the weak-measurement method (WMM). (c) is the OTOC evaluated using the irreversibility-susceptibility method (ISM), which shows a larger standard deviation because of its reliance on the weak interactions.